Sharp stability of a string with local degenerate Kelvin–Voigt damping

Abstract

AbstractThis paper is on the asymptotic behavior of the elastic string equation with localized degenerate Kelvin–Voigt damping where on , and on for . It is known that the optimal decay rate of solution is in the limit case and exponential for . When , the damping coefficient is continuous, but its derivative has a singularity at the interface . In this case, the best known decay rate is , which fails to match the optimal one at . In this paper, we obtain a sharper polynomial decay rate . More significantly, it is consistent with the optimal polynomial decay rate at and uniform boundedness of the resolvent operator on the imaginary axis at (consequently, the exponential decay rate at as ). This is a big step toward the goal of obtaining eventually the optimal decay rate.